Let us not impose any IR cutoff on Minkowski space. The massive scalar's vacuum is given by a normalizable state. However, since there is a continuum of energies for the massless scalar starting from the vacuum itself, is it correct to conclude that the massless scalar's vacuum is non-normalizable? (Note that the continuum of energy states for massive scalar doesn't start from the vacuum)
There might be another way to see this issue. A free particle in non-relativistic QM doesn't have a normalizable vacuum, however as soon as you turn on a potential we get a normalizable vacuum. Does this logic extend to massless scalar's case as well?